It's well known that BB(n) is not a computable function. Your
question is not the typical phrasing of the Busy Beaver problem, so
I'm not sure how it relates to it. I don't think they're equivalent,
as BB is about a certain kind of Turing machine, and you're asking
about finite automata.
The enumeration of all finite state automata of n states (for some
finite alphabet) should be computable, but I don't think it's a
tractable problem. Also, all finite automata will halt on a finite
input string, as they consume one input token per state transition.
Now, if you actually meant automata in general rather than finite
automata, that's another story, and I haven't put any thought into
that one yet.
--Levi
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